Algebraic Approach to Specifying Part-Whole Associations

Introduction Expressive power and deductive power are two critical characteristics of knowledge representation languages. They capture respectively, what information can be explicitly stated, and what information can be deduced. Object (class) based representations have gained almost universal acceptance because of their ability to capture inter-class associations and their implied ability to reason about these associations. While this is true for taxonomical relations (generalizations, specializations), this is far from being true for structural associations relating a whole to its parts. Most graphical and formal languages provide constructs for stating part-whole associations, but most languages have limited or no support for making inferences from them. This shortcoming is not a new revelation. Extensive research has been ongoing in philosophy, linguistics, logic, artificial intelligence, and software engineering with a focus to formalize the semantics of the part-whole association. This research resulted in a diverse pool of formalisms, some deemed too weak and thus not very useful, and some deemed too strong and thus not very usable; and most deemed both too weak and too strong because they do not capture all the properties of interest to some application domain, and capture properties that do not hold in the same application domain. This paradoxical state of affairs is in fact a reflection of the nature of the part-whole association. While there is an intuitive universal understanding of what the association means, the specific properties that one needs to reason about vary from one domain to the next, and from one application to the next. In this paper, we take an approach to defining part-whole associations that account both for the universality and the variability: • We account for the “universality” by defining all partwhole associations in terms of a common set of primitive associations. • We account for the “variability” by the fact that each part-whole association may be a different combination of primitive associations. We define an algebra of associations that serves as a basis for the deductive power of languages capturing the part-whole association. Most researchers focusing on the representation of partwhole relations base their work on Description Logics (DL) [ 1]. We use a Tarski-like algebra of binary relations which has a similar expressive power but presents the convenience of an algebra. A relation on a set Σ is a subset of Σ ×Σ. Constant relations on set Σ include: the Universal relation L=Σ×Σ, the identity relation I={(s,s)|s ∈ Σ} and the empty relation Φ={}. Given A, a subset of Σ, we define I(A) as {(s,s)| s∈A}. In addition, given two sets A and B subsets of Σ, we define the relation D(A,B)=A×B={(s,s’)| s∈A and s’∈ B}. The intersection of two relations R and R’ is defined by: The composition of two relations R and R’ is denoted by R◦R’ and defined by: R◦R’ ={(s,s’)| ∃ s”: (s,s”) ∈ R and (s”,s’) ∈ R’}. The inverse of a relation R is denoted by R and defined by R ={(s,s’)| (s’.s) s∈R}. The nucleus of a relation R is denoted by ν(R) and defined by ν(R) = RoR. The co-nucleus of a relation R is denoted by γ(R) and defined by γ(R) = RoR.